Confidence Intervals for Proportions

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Chapter 19

• Confidence Intervals for Proportions

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First Standard Error (Ch 18)

• Do we normally know the true population mean?
• What can we use in place of p, if we dont know
  it? (since we need it for the standard deviation
  of the sampling distribution!)

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Standard Error

• We can use phat, the sample proportion.
• Slightly different quantity, new name
• Standard Error

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Statistical Inference

• Two most common types of stat. Inference
• Confidence Intervals, to estimate the value of a
  population paramter
• Hypothesis Tests, to assess evidence for a claim
  about a population.
• Today Confidence Intervals

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Election Polls

• Recall the Perdue/Barnes race for governor
• Pollsters predicted Barnes would win right up
  until the election
• Perdue won, though. How could the polls have been
  that wrong?
• Sample Poll results (made up)
• favoring Barnes 49
• favoring Perdue 44
• /- 3
• Clear winner by 5, right?

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Sample proportions and p

• The pollsters found sample proportions for their
  samples.
• Are these exactly the same as the true
  proportion, p?
• The pollsters know theyre likely off from the
  true p by a bit, so they hedge their bets with
  a Margin of Error. (the /-)

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Sampling Distributions and CIs

• Our knowledge of sampling distributions lets us
  estimate how close we are likely to be to the
  true p.
• Lets take a look at how this works

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Example Problem 12

• A Gallup poll found that 8 of a random sample of
  1012 adults approve of attempts to clone a human.
• Given phat 0.08 n1012
• Is true p 0.08? Not likely!
• How variable is the distribution of phats?
• Since we have only phat, must use SE(phat) to
  estimate
• We also know, from the Normal Model
• About 95 of ALL samples will have phat within 2
  SE of true p.
• (ie, phat /- (20.008528) lt p

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12 continued

• So, the odds are good that were not that far
  away from the true proportion, p!
• Phat p /- 2SE(phat)
• This means, also, that p is likely within 2SE of
  phat!! (just rearrange the equation!)
• Ie, 95 of all samples will have a sample
  proportion, phat, within 2 Standard Errors of the
  true proportion, p.
• Important point Does p move from sample to
  sample?
• No! It is the population parameter- fixed, but
  generally unknown.
• Phat does vary with each sample, but p does NOT.

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Confidence

• So, when you take a sample and find phat, how
  sure are you that you are close to p?
• This is our confidence level.
• We can say we are 95 confident that the true
  proportion p lies between phat-2SE and phat2SE
• Two possibilities with any sample
• Our interval captures the true proportion
• Our SRS is one of the few samples for which phat
  is more than 2 SE away from p, ie a bad sample.
• We can NEVER know for certain whether our sample
  is a good one or not!!

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Form for CI

• Formal name One-proportion z-interval
• Or, Confidence interval for a proportion
• General Form for ANY CI
• Estimate /- Margin of Error
• Estimate our best guess for p, which is phat.
• ME Shows how accurate we think our guess is,
  based on the variability of the sample.
• A 95 CI catches p in 95 of all samples

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Other Confidence Levels

• Any CI has 2 (required) parts
• An interval, computed from the data
• A Confidence level, giving the probability that
  the method produces an interval containing
  parameter p.
• We are NOT limited to using 95 confidence all
  the time, though!
• You can chose the confidence level.
• Usually 90 or higher, since we want to be
  certain.
• Its really a balancing act between precision
  (the size of the ME) and certainty (Confidence
  level).

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Balancing Confidence Level and ME

• If I wanted to be 100 confident about something,
  Id need to use a HUGE interval (too large to be
  useful)
• Ex Im 100 confident that between 0 and 100
  of people will vote for Bush in the next
  election.
• We used 2SE in our CI based on the 68-95-99.7
  rule. (for 95)
• What would we use for 99.7 confidence? (approx)

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Z

• We can use any confidence level by using the
  appropriate z-score, called the Critical value,
  z.
• Using the Normal table, we can find that a z of
  1.96 captures exactly the middle 95 of all
  sample proportions. (2 was just an approximation)
• What z would capture the middle 90 exactly?
• Look up proportion 0.05 ? gives a z 1.645

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More zs

• Another common CI level is 99
• Common zs (you should know these!)
• 90 z 1.645
• 95 z 1.96
• 99 z2.576
• Can find z for any confidence level, though,
  including 80, 83.4, etc.

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The Confidence Interval

• The confidence Interval for p is given by
• Phat /- z SE(phat)
• Where z is determined from the confidence level,
  C
• And
• Assumptions and conditions must be met for CI to
  be valid
• Assume
• Independent values
• Random Sample
• Sample lt 10 of population
• Condition (check)
• Sample big enough- at least 10 successes, 10
  failures

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12 Cloning poll continued

• Lets check our cloning poll conditions and
  assumptions
• Independence Gallop phoned random sample,
  unlikely to be dependent.
• Random yes (can generally trust Gallop methods)
• 10 1012 ltlt 10 all adults
• Big enough 8(1012)80.96, 92(1012)931.04
• Now, lets find the ME for 95 confidence
• Margin of error is the /- part, zSE(phat)
• ME

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Finish 12 Conclusion

• Therefore, We are 95 confident that the true
  proportion of adults who approve of cloning
  humans is within 1.67 of 8.
• (or, .is between 6.33 and 9.67)
• When stating a confidence interval, your best
  final answer is a sentence (such as)
• We are ___ confident that the true proportion
  _______ is between ___ and ____.

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ME, n and C

• What will happen to the ME we calculated if we
  lower the confidence level (C) to 90? Why?
• Find the CI for 12 for 90 confidence
• ME1.645SE(phat) 0.014028
• CI 0.08/- 0.014 ?(6.6, 9.4)

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Making ME smaller

• Ideally, we want both high confidence AND a small
  ME.
• High confidence method almost always gives
  correct results
• Small ME weve pinned down p precisely.
• How can we make ME smaller?
• Two choices
• Decrease the confidence level? decreases z
• Increase the sample size ? decreases SE(phat)
• This is the better option, but more expensive.

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How big a sample do we need?

• If you have a desired ME and confidence level,
  you can determine how large a sample is needed.
• Solve for n!
• We can do this, since we know z, ME, phat, qhat
• Same equation
• Say we want to be within 1 of p for cloning
  poll, with 95 confidence.
• ? n2827.41.
• We MUST round this UP
• to 2828 people. (why?)

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Example 18

• 21 of 110 surveyed local middle schoolers
  reported having been drunk.
• What is phat?
• Estimate the true proportion of drunk
  middle-schoolers, with 95 confidence.
• Phat /- z SE(phat)

We are 95 confident that between 11.7 and 26.4
of middle schoolers have been drunk.

• If the national is 30, are the local students
  actually less likely to have been drunk?
• Yes- 30 is above our CI

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18 Changing n and C

• If we make our sample larger, will the CI get
  larger or smaller?
• If we lower our confidence level, will the CI get
  larger or smaller?

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Summary of new formulas

• Standard Error, SE(phat)
• General CI formula phat /- ME
• CI formula for proportions
• phat /- z SE(phat)
• How big a sample needed